177 research outputs found
Degenerating Black Saturns
We investigate the possibility of constructing degenerate Black Saturns in
the family of solutions of Elvang-Figueras. We demonstrate that such solutions
suffer from naked singularities.Comment: 14 LaTeX page
On Projections in the Noncommutative 2-Torus Algebra
We investigate a set of functional equations defining a projection in the
noncommutative 2-torus algebra . The exact solutions of these
provide various generalisations of the Powers-Rieffel projection. By
identifying the corresponding classes we get an insight into
the structure of projections in
An algebraic formulation of causality for noncommutative geometry
We propose an algebraic formulation of the notion of causality for spectral
triples corresponding to globally hyperbolic manifolds with a well defined
noncommutative generalization. The causality is given by a specific cone of
Hermitian elements respecting an algebraic condition based on the Dirac
operator and a fundamental symmetry. We prove that in the commutative case the
usual notion of causality is recovered. We show that, when the dimension of the
manifold is even, the result can be extended in order to have an algebraic
constraint suitable for a Lorentzian distance formula.Comment: 24 pages, minor changes from v2, to appear in Classical and Quantum
Gravit
Asymptotic and exact expansions of heat traces
We study heat traces associated with positive unbounded operators with
compact inverses. With the help of the inverse Mellin transform we derive
necessary conditions for the existence of a short time asymptotic expansion.
The conditions are formulated in terms of the meromorphic extension of the
associated spectral zeta-functions and proven to be verified for a large class
of operators. We also address the problem of convergence of the obtained
asymptotic expansions. General results are illustrated with a number of
explicit examples.Comment: 44 LaTeX pages, 2 figure
Spectral dimensions and dimension spectra of quantum spacetimes
Different approaches to quantum gravity generally predict that the dimension
of spacetime at the fundamental level is not 4. The principal tool to measure
how the dimension changes between the IR and UV scales of the theory is the
spectral dimension. On the other hand, the noncommutative-geometric perspective
suggests that quantum spacetimes ought to be characterised by a discrete
complex set -- the dimension spectrum. Here we show that these two notions
complement each other and the dimension spectrum is very useful in unravelling
the UV behaviour of the spectral dimension. We perform an extended analysis
highlighting the trouble spots and illustrate the general results with two
concrete examples: the quantum sphere and the -Minkowski spacetime, for
a few different Laplacians. In particular, we find out that the spectral
dimensions of the former exhibit log-periodic oscillations, the amplitude of
which decays rapidly as the deformation parameter tends to the classical value.
In contrast, no such oscillations occur for either of the three considered
Laplacians on the -Minkowski spacetime.Comment: 35 pages, 7 figures, v2 some comments and references added, summary
extended, title change
Noncommutative geometry, Lorentzian structures and causality
The theory of noncommutative geometry provides an interesting mathematical
background for developing new physical models. In particular, it allows one to
describe the classical Standard Model coupled to Euclidean gravity. However,
noncommutative geometry has mainly been developed using the Euclidean
signature, and the typical Lorentzian aspects of space-time, the causal
structure in particular, are not taken into account. We present an extension of
noncommutative geometry \`a la Connes suitable the for accommodation of
Lorentzian structures. In this context, we show that it is possible to recover
the notion of causality from purely algebraic data. We explore the causal
structure of a simple toy model based on an almost commutative geometry and we
show that the coupling between the space-time and an internal noncommutative
space establishes a new `speed of light constraint'.Comment: 24 pages, review article. in `Mathematical Structures of the
Universe', eds. M. Eckstein, M. Heller, S.J. Szybka, CCPress 201
Causality in noncommutative two-sheeted space-times
We investigate the causal structure of two-sheeted space-times using the
tools of Lorentzian spectral triples. We show that the noncommutative geometry
of these spaces allows for causal relations between the two sheets. The
computation is given in details when the sheet is a 2- or 4-dimensional
globally hyperbolic spin manifold. The conclusions are then generalised to a
point-dependent distance between the two sheets resulting from the fluctuations
of the Dirac operator.Comment: 26 pages, 2 figure
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